Difference b/w Combinations of random variables and transforms of rv's.

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Sidhant Shivram
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Hey there (:

Can someone briefly explain to me the difference b/w combinations of random variables and transforms of random variables (as far as I know, only "combinations of rv's" is included interest Edexcel's A Level Math S4 module but I'm just curious).

Also, the topic is explained as : "distribution of linear combinations of independent normal random variables". So I was wondering, why is "linear" mentioned there? What is 'linear' about the distribution of the said combinations of independent normal rv's?

Thank you very much! (:

PS : I haven't given this much thought & I won't be able to research about it either since I'm running out of time (will be travelling soon) and I need to clarify my doubts faster. Hence, the post here, on TSR. Under normal circumstances, I'd have tried to find out for myself by researching about it online and post on TSR only if I wasn't able to get answers even then)
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Farhan.Hanif93
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(Original post by Sidhant Shivram)
Hey there (:

Can someone briefly explain to me the difference b/w combinations of random variables and transforms of random variables (as far as I know, only "combinations of rv's" is included interest Edexcel's A Level Math S4 module but I'm just curious).
Roughly speaking, a transformation of random variables is given by some function of the random variables, such as \sin (X+Y) or 2|X|+\dfrac{1}{Y} +\sqrt{5Z} etc etc, where X,Y,Z are suitable R.V.s.

I've never heard the phrase "combination of r.v.s" but I'd say it's reasonable to assume that it is the same thing as a transformation. A linear combination, on the other hand, is a more specific type of transformation.

Also, the topic is explained as : "distribution of linear combinations of independent normal random variables". So I was wondering, why is "linear" mentioned there? What is 'linear' about the distribution of the said combinations of independent normal rv's?
"Linear" is used to describe the type of transformation that yields a "linear combination" of R.V.s, as opposed to "linearity" of it's distribution (whatever that means).

A transformation X of random variables X_1, …, X_n is called linear if X = a_1X_1 + a_2 X_2 + … + a_nX_n, where a_i are suitably chosen constants, and we say that X is a linear combination of the X_i.

It's called linear because the random variables X_1, … , X_n appear in a linear fashion (in the same way that y=mx+c represents a straight line in the x-y plane - the X_i are not transformed themselves i.e. aX_1^2 + bX_2 is not a linear transformation because X_1 has been transformed in a non-linear fashion.)
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Sidhant Shivram
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(Original post by Farhan.Hanif93)
Roughly speaking, a transformation of random variables is given by some function of the random variables, such as \sin (X+Y) or 2|X|+\dfrac{1}{Y} +\sqrt{5Z} etc etc, where X,Y,Z are suitable R.V.s.

I've never heard the phrase "combination of r.v.s" but I'd say it's reasonable to assume that it is the same thing as a transformation. A linear combination, on the other hand, is a more specific type of transformation.


"Linear" is used to describe the type of transformation that yields a "linear combination" of R.V.s, as opposed to "linearity" of it's distribution (whatever that means).

A transformation X of random variables X_1, …, X_n is called linear if X = a_1X_1 + a_2 X_2 + … + a_nX_n, where a_i are suitably chosen constants, and we say that X is a linear combination of the X_i.

It's called linear because the random variables X_1, … , X_n appear in a linear fashion (in the same way that y=mx+c represents a straight line in the x-y plane - the X_i are not transformed themselves i.e. aX_1^2 + bX_2 is not a linear transformation because X_1 has been transformed in a non-linear fashion.)
Oh okay! That makes sense! Thank you so much! (:


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